While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure
μ
\mu
is the countable dense subset
{
μ
(
U
)
:
U
\{ \mu (U) : U
is clopen
}
\}
of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure
μ
\mu
is good if whenever
U
,
V
U, V
are clopen sets with
μ
(
U
)
>
μ
(
V
)
\mu (U) > \mu (V)
, there exists
W
W
a clopen subset of
V
V
such that
μ
(
W
)
=
μ
(
U
)
\mu (W) = \mu (U)
. These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense,
G
δ
G_{\delta }
conjugacy class.